Nonparametric warped kernel estimators
Résumé
In this work, we propose a general method of adaptive nonparametric estimation, based on warped kernels. The aim is to estimate a real-valued function $s$ from a sample of random variables $(X,Y)$. We first deal with the auxiliary function $g=s\circ \phi_X$, for a bijective map $\phi_X$ depending on the distribution of the variable $X$: we consider a collection of kernel estimates built with the warped data $(\phi_X(X),Y)$. The data-driven selection of the best bandwidth is done with a model selection device in the spirit of Goldenshluger and Lepski~(2011). This leads to an estimator $\hat{g}$ for the function $g$, which is then warped to estimate the target function $s$ by $\hat{s}=\hat{g}\circ \hat{\phi}_X$ where $\hat{\phi}_X$ is an estimate for $\phi_X$. The interest is twofold. From the practical point of view, the estimator can be computed easily and fastly, thanks to its simple explicit expression. From the theoretical point of view, the squared-bias/variance trade-off is realized: we derive non-asymptotic risk bounds. This general method permits to handle various problems such as additive and multiplicative regression, conditional density estimation, hazard rate estimation based on randomly right censored data, and cumulative distribution function estimation from current-status data.
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